L-functions and theta correspondence for classical groups

Using the doubling method of Piatetski-Shapiro and Rallis, we develop a theory of local factors of representations of classical groups and apply it to give a necessary and sufficient condition for nonvanishing of global theta liftings in terms of analytic properties of the L-functions and local theta correspondence.     

On correspondences between certain automorphic forms on Sp_{4n} (\mathbb{A}) and \widetilde{Sp}_{2n} (\mathbb{A})

In 2005, Ginzburg, Rallis and Soudry constructed, in terms of residues of certain Eisenstein series, and by use of the descent method, families of nontempered automorphic representations of $Sp_{4nm} (\mathbb{A})$ and $\widetilde{Sp}_{2n(2m - 1)} (\mathbb{A})$ , which generalized the classical work of Piatetski-Shapiro on Saito-Kurokawa liftings. In this paper, we introduce a new framework (Diagrams of Constructions) in order to establish explicit relations among the representations introduced in [GRS05]. In particular, we prove that these constructions yield bijections between a certain set of cuspidal automorphic forms on $\widetilde{Sp}_{2n} (\mathbb{A})$ and a certain set of square-integrable automorphic forms of $Sp_{4n} (\mathbb{A})$ . The proofs use new interpretations of composition of two consecutive descents with explicit identities, which we expect to be very useful to further investigation of the automorphic discrete spectrum of classical groups.     

Epipelagic representations and rigid local systems

We construct automorphic representations for quasi-split groups G over the function field \(F=k(t)\) one of whose local components is an epipelagic representation in the sense of Reeder and Yu. We also construct the attached Galois representations under the Langlands correspondence. These Galois representations give new classes of conjecturally rigid, wildly ramified \({}^{L}{G}\)-local systems over \(\mathbb {P}^{1}-\{0,\infty \}\) that generalize the Kloosterman sheaves constructed earlier by Heinloth, Ngô and the author. We study the monodromy of these local systems and compute all examples when G is a classical group.     

On symplectic periods of the discrete spectrum ofGL 2n

We provide a formula for the symplectic period of an automorphic form in the discrete spectrum ofGL _2n.It is a generalization of a formula of Jacquet and Rallis.     

Moment sets and unitary dual for the diamond group

The diamond group G is a solvable group, semi-direct product of R with a (2n+1)-dimensional Heisenberg group Hn. We consider this group as a first example of a semi-direct product with the form R?N where N is nilpotent, connected and simply connected.Computing the moment sets for G, we prove that they separate the coadjoint orbits and its generic unitary irreducible representations.Then we look for the separation of all irreducible representations. First, moment sets separate representations for a quotient group G⁻ of G by a discrete subgroup, then we can extend G to an overgroup G⁺, extend simultaneously each unitary irreducible representation of G to G⁺ and separate the representations of G by moment sets for G⁺.     

Central Pairs, Galois Theory and Automorphic Forms

A central pair over a field k of characteristic 0 consists of a finite Abelian group which is equipped with a central 2-cocycle with values in the multiplicative group k ^* of k. In this paper we use specific central pairs to construct a class of projective representations of the absolute Galois group G _k of k and if k is a number field we investigate the liftings of these projective representations to linear representations of G _k . In particular we relate these linear representations to automorphic representations. It turns out that some of these automorphic representations correspond to certain indefinite modular forms already constructed by E. Hecke.     

Bessel functions for GL 2

In classical analytic number theory there are several trace formulas or summation formulas for modular forms that involve integral transformations of test functions against classical Bessel functions. Two prominent such are the Kuznetsov trace formula and the Voronoi summation formula. With the paradigm shift from classical automorphic forms to automorphic representations, one is led to ask whether the Bessel functions that arise in the classical summation formulas have a representation theoretic interpretation. We introduce Bessel functions for representations of GL ₂ over a finite field first to develop their formal properties and introduce the idea that the γ-factor that appears in local functional equations for L-functions should be the Mellin transform of a Bessel function. We then proceed to Bessel functions for representations of GL ₂(?) and explain their occurrence in the Voronoi summation formula from this point of view. We briefly discuss Bessel functions for GL ₂ over a p-adic field and the relation between γ-factors and Bessel functions in that context. We conclude with a brief discussion of Bessel functions for other groups and their application to the question of stability of γ-factors under highly ramified twists.     

Computing the Automorphic Chromatic Index of Certain Snarks

The automorphic G-chromatic index of a graph Γ is the minimum integer m for which Γ has a proper edge-coloring with m colors which is preserved by the full automorphism group G of Γ. We determine the automorphic G-chromatic index of each member of four infinite classes of snarks: type I Blanu?a snarks, type II Blanu?a snarks, Flower snarks and Goldberg snarks.     

Automorphic Plancherel density theorem

Let F be a totally real field, G a connected reductive group over F, and S a finite set of finite places of F. Assume that G(F ?_? ?) has a discrete series representation. Building upon work of Sauvageot, Serre, Conrey-Duke-Farmer and others, we prove that the S-components of cuspidal automorphic representations of $G\left( {\mathbb{A}_F } \right)$ are equidistributed with respect to the Plancherel measure on the unitary dual of G(F _S ) in an appropriate sense. A few applications are given, such as the limit multiplicity formula for local representations in the global cuspidal spectrum and a quite flexible existence theorem for cuspidal automorphic representations with prescribed local properties. When F is not a totally real field or G(F ?_? ?) has no discrete series, we present a weaker version of the above results.     

On Atkin and Swinnerton-Dyer congruence relations (3)

It is now well known that Hecke operators defined classically act trivially on genuine cuspforms for noncongruence subgroups of SL₂(Z). Atkin and Swinnerton-Dyer speculated the existence of p-adic Hecke operators so that the Fourier coefficients of their eigenfunctions satisfy three-term congruence recursions. In the previous two papers with the same title ([W.C. Li, L. Long, Z. Yang, On Atkin and Swinnerton-Dyer congruence relations, J. Number Theory 113 (1) (2005) 117-148] by W.C. Li, L. Long, Z. Yang and [A.O.L. Atkin, W.C. Li, L. Long, On Atkin and Swinnerton-Dyer congruence relations (2), Math. Ann. 340 (2) (2008) 335-358] by A.O.L. Atkin, W.C. Li, L. Long), the authors have studied two exceptional spaces of noncongruence cuspforms where almost all p-adic Hecke operators can be diagonalized simultaneously or semi-simultaneously. Moreover, it is shown that the l-adic Scholl representations attached to these spaces are modular in the sense that they are isomorphic, up to semisimplification, to the l-adic representations arising from classical automorphic forms.In this paper, we study an infinite family of spaces of noncongruence cuspforms (which includes the cases in [W.C. Li, L. Long, Z. Yang, On Atkin and Swinnerton-Dyer congruence relations, J. Number Theory 113 (1) (2005) 117-148; A.O.L. Atkin, W.C. Li, L. Long, On Atkin and Swinnerton-Dyer congruence relations (2), Math. Ann. 340 (2) (2008) 335-358]) under a general setting. It is shown that for each space in this family there exists a fixed basis so that the Fourier coefficients of each basis element satisfy certain weaker three-term congruence recursions. For a new case in this family, we will exhibit that the attached l-adic Scholl representations are modular and the p-adic Hecke operators can be diagonalized semi-simultaneously.     

On the standard modules conjecture

Let G be a quasi-split p-adic group. Under the assumption that the local coefficients C _ψ defined with respect to ψ-generic tempered representations of standard Levi subgroups of G are regular in the negative Weyl chamber, we show that the standard module conjecture is true, which means that the Langlands quotient of a standard module is generic if and only if the standard module is irreducible.     

The game of arboricity

Using a fixed set of colors C, Ann and Ben color the edges of a graph G so that no monochromatic cycle may appear. Ann wins if all edges of G have been colored, while Ben wins if completing a coloring is not possible. The minimum size of C for which Ann has a winning strategy is called the game arboricity of G, denoted by A_g(G). We prove that A_g(G)?3k for any graph G of arboricity k, and that there are graphs such that A_g(G)?2k-2. The upper bound is achieved by a suitable version of the activation strategy, used earlier for the vertex coloring game. We also provide two other strategies based on induction and acyclic colorings.     

An analogue of the BGG resolution for locally analytic principal series

Let G be a connected reductive quasi-split algebraic group over a field L which is a finite extension of the p-adic numbers. We construct an exact sequence modelled on (the dual of) the BGG resolution involving locally analytic principal series representations for G(L). This leads to an exact sequence involving spaces of overconvergent p-adic automorphic forms for certain groups compact modulo centre at infinity.     

On local newforms for unramified U(2, 1)

Let G be the unramified unitary group in three variables defined over a p-adic field F with p ≠ 2. In this paper, we investigate local newforms for irreducible admissible representations of G. We introduce a family of open compact subgroups {K _n } _n≥0 of G to define the local newforms for representations of G as the K _n -fixed vectors. We prove the existence of local newforms for generic representations and the multiplicity one property of the local newforms for admissible representations.     

Correspondance de Jacquet-Langlands pour les corps locaux de caractéristique non nulle

In this article we prove the Jacquet-Langlands local correspondence in non-zero characteristic. Let F be a local field of non-zero charactersitic and G′ an inner form of GL_n(F); then, following [17], we prove relations between the representation theory of G′ and the representation theory of an inner form of GL_n(L), where L is a local field of zero characteristic close to F. The proof of the Jacquet-Langlands correspondence between G′ and GL_n(F) is done using the above results and ideas from the proof by Deligne, Kazhdan and Vignéras [10] of the zero characteristic case. We also get the following, already known in zero characteristic: orthogonality relations for G′, inequality involving conductor and level for representations of G′ and finiteness for automorphic cuspidal representations with fixed component at almost every place for an inner form of GL_n over a global field of non-zero characteristic.     

Zeta functions of prehomogeneous vector spaces with coefficients related to periods of automorphic forms

The theory of zeta functions associated with prehomogeneous vector spaces (p.v. for short) provides us a unified approach to functional equations of a large class of zeta functions. However the general theory does not include zeta functions related to automorphic forms such as the HeckeL-functions and the standardL-functions of automorphic forms on GL(n), even though they can naturally be considered to be associated with p.v.’s. Our aim is to generalize the theory to zeta functions whose coefficients involve periods of automorphic forms, which include the zeta functions mentioned above. In this paper, we generalize the theory to p.v.’s with symmetric structure ofK _ε-type and prove the functional equation of zeta functions attached to automorphic forms with generic infinitesimal character. In another paper, we have studied the case where automorphic forms are given by matrix coefficients of irreducible unitary representations of compact groups. Dedicated to the memory of Professor K G Ramanathan     

On the Minimum Number of Spanning Trees in k‐Edge‐Connected Graphs

We show that a k‐edge‐connected graph on n vertices has at least spanning trees. This bound is tight if k is even and the extremal graph is the n‐cycle with edge multiplicities . For k odd, however, there is a lower bound , where . Specifically, and . Not surprisingly, c₃ is smaller than the corresponding number for 4‐edge‐connected graphs. Examples show that . However, we have no examples of 5‐edge‐connected graphs with fewer spanning trees than the n‐cycle with all edge multiplicities (except one) equal to 3, which is almost 6‐regular. We have no examples of 5‐regular 5‐edge‐connected graphs with fewer than spanning trees, which is more than the corresponding number for 6‐regular 6‐edge‐connected graphs. The analogous surprising phenomenon occurs for each higher odd edge connectivity and regularity.